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Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question

We consider the following two classes of smooth maps on $\mathbb{R}^{n}.$ ( $n$ is not necessarily even):

$HP_{1}$: $\;$A smooth map $f:\mathbb{R}^{n} \to \mathbb{R^{n}}$ satisfies property $HP_{1}$ if the pull back metric $f^{*}(g)$ is a conformal metric for every conformal metric $g$ on $\mathbb{R}^{n}.$ ( By conformality I mean $g=e^{h} \sum dx_{i}^{2})$

$HP_{2}$: $\;$A smooth map $X$ on $\mathbb{R}^{n}$ with coordinates $X=(P_{1},P_{2},\ldots,P_{n})$ satisfies $HP_{2}$ if the space of harmonic functions is invariant under the derivational operator $U \mapsto X.U=\sum P_{i} \partial U/ \partial x_{i}$

Questions:

  1. Assume $n=2k$, identify $\mathbb{R}^{2k}$ with $\mathbb{C}^{k}$. Let $f$ satisfies $HP_{1}$. Is $f$ a holomorphic map on $\mathbb{C}^{k}$? The same question for $HP_{2}$?

  2. Are the above two properties, equivalents?(A map is $HP_{1}$ iff it is $HP_{2}$?)

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Why does your property hold for maps C^2 \to C^2 ? Also see mathoverflow.net/questions/93069/…;. –  user36931 Apr 6 at 13:14
    
By the above i meant property one. I see it if the map is "diagonal" but not if it mixes up the two factors. –  user36931 Apr 6 at 13:26
    
@user36931 However I wrote in the "motivation" "HP_{2} implies holomorphicity", but now I realize that my argument was not complete! What is your reason for "diagonal maps"?(Note that HP2 was that "harmonic maps are invariant under derivation of X" th question is X holomorphic? –  Ali Taghavi Apr 7 at 4:13

1 Answer 1

This is an extended comment. The first property $HP_1$ simply says that $f$ is conformal. It is not true that holomorphic maps of $C^n=R^{2n}$ are conformal, except when $n=1$. In fact there are very few conformal maps in $R^n$ for $n\geq 3$: they are only Mobius transfomrations (compositions of reflections).

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Moreover conformal maps need not be holomorphic (even rotations can send a complex line to a totally real plane). –  Benoît Kloeckner Apr 6 at 18:56

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